Pondering Productivity answer explained:
(a) Eight of Smith’s hens will lay 7 × (7/6) eggs in 6/5 days for a rate of (7/8) × (7/6) ÷ (6/5) = 245/288 eggs per hen per day. Six of Jones’s hens will lay 5 × (7/6) eggs in 8/7 days for a rate of (5/6) × (7/6) ÷ (8/7) = 245/288 eggs per hen per day. The hens from both farms are equally productive.
(b) With 48 hens Smith gets 245/6 eggs daily so must wait 6 days to get 245 eggs.
(c) With 300 hens Jones gets 6125/24 eggs daily so must wait 24 days to get 6125 eggs.

Cornfield Planning answer explained:
Cut out quarter circles on the corners as shown in the figure. The area of such a shape is A = 10000 – 4r2 + πr2. The perimeter of the shape is P = 400 – 8r + 2πr. The ratio A/P takes on a maximum value of 100/(2+√π) = 26.5079… when r = 100/(2+√π) = 26.5079… If the cornfield were either a circle or a square, the ratio A/P would be 25.